Description:

Topics in algebraic stacks and how it relates to K-stability and other moduli notions. We plan to focus on topics including, but not limited to, foliations, algebraic stacks, K-stability and K-moduli, moduli theories and birational geometry. We plan for this seminar to be a supplement to the usual weekly algebraic geometry seminar, covering a broad range of topics in moduli theories which have been prevalent in wide areas of modern algebraic geometry.

Speaker: Ruadhai Dervan

Affiliation: University of Warwick

Time: Wednesday, January 14th, 15:30-16:30

Venue: Lecture Hall B725, Shuangqing Complex Building A (清华大学双清综合楼A座B725报告厅) (In Person)

Zoom Meeting ID: 262 865 5007 Passcode: YMSC

Title: Metric wall-crossing

Abstract: When a reductive group acts on a projective variety, a choice of (linearised) ample line bundle gives a choice of quotient. Wall-crossing (or VGIT) explains how the quotient space changes with the choice of line bundle: the quotients vary birationally, by flips, and only finitely finite birational models can occur.

I will describe a (Kähler) metric version of these results. Each quotient admits a natural choice of Kähler metric, through a symplectic quotient construction. I will prove metric convergence (towards walls), and the existence of metric flips (across walls), when one suitably varies the choice of line bundle determining the quotient. I will use these results to motivate analogous conjectures governing the metric geometry of moduli spaces in wall-crossing problems in algebraic geometry.

Oct. 22, 2025

Speaker: Iacopo Brivio

Affiliation: Center for Mathematical Sciences and Applications (CMSA), Harvard University

Title: Antiitaka conjecture for fibrations in positive characteristic.

Abstract: A famous conjecture by Iitaka predicts that, given a fibration $f\colon X\to Y$ of smooth complex varieties with general fiber $F$, then the inequality $\kappa(K_X)\geq \kappa(K_F)+\kappa(K_Y)$ holds. It was recently shown by Chang that, when the stable base locus $\mathbb{B}(-K_X)$ is $f$-vertical, then a similar inequality holds for the anticanonical divisor: $\kappa(-K_X)\leq \kappa(-K_F)+\kappa(-K_Y)$. Over fields of positive characteristic it is known that both Iitaka's conjecture and Chang's theorem can fail. However the expectation is that, when $F$ is sufficiently well-behaved with respect to the Frobenius morphism, then these results should hold. In this talk I will show that this is the case for Chang's theorem. This is based on joint work with Benozzo and Chang.